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2394 bytes, as of the version from 2026-10-27 04:54 (current). Recorded here, not run.
# Sources and conventions for T538
1. A. Dilek Güngör and Ş. Burcu Bozkurt, *On the Distance Estrada Index of
Graphs*, Hacettepe Journal of Mathematics and Statistics 38(3) (2009),
277–283. https://dergipark.org.tr/en/download/article-file/86791
Read pp. 277–279: graph and distance convention, Definition 2.1 (8),
moment expansion (9), and Lemma 2.2. Stored eigenvalues retain their signs
and algebraic multiplicities. There is no centering, division by vertex
count, or exponential scale parameter.
2. Ernesto Estrada, *The many facets of the Estrada indices of graphs and
networks*, SeMA Journal 79 (2022), 57–125,
doi:10.1007/s40324-021-00275-w. Definition 23, equation (16.6), independently
confirms the shortest-path matrix and trace-of-exponential definition.
https://link.springer.com/article/10.1007/s40324-021-00275-w
3. Brendan McKay, connected graph catalogues:
https://users.cecs.anu.edu.au/~bdm/data/graphs.html
Read the Simple graphs counts: 1, 2, 6, 21, 112, 853 for orders 2–7.
Format specification:
https://users.cecs.anu.edu.au/~bdm/data/formats.txt
Read the graph6 edge order and encoding. The independent checker decodes
graph6 directly and compares with Sage's distance matrix.
4. T237, T238, T536 and T537 were read from the live authenticated NumberDB
API before building. `comparison-tables.json` preserves the relevant
entries. T536 and T537 are sibling drafts at build time. Stable T-number
links are used because the table-document API does not return slugs and
a draft page cannot be fetched through the public page route with a bearer
key. T-number HREFs are explicitly supported by NumberDB's link resolver.
The complete-graph formula follows from the spectrum n-1, -1 (n-1 times).
For K(a,b), the two-part quotient matrix has eigenvalues
(a+b-2) +/- sqrt(a*a-a*b+b*b), and the remaining a+b-2 eigenvalues are -2.
These decompositions prove the formulas for the full domains; the numerical
checks exercise every applicable selected graph. Jensen's inequality and
trace(D)=0 give DEE(G)>n for n>=2. The largest-eigenvalue bounds follow from
positivity of every exponential term.
The archived coordinated skill manifest is
`agents/runs/20261027T044107Z-build-skills/manifest.json`.
Its local and live files have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.